Properties

 Label 4-21e2-1.1-c2e2-0-4 Degree $4$ Conductor $441$ Sign $1$ Analytic cond. $0.327422$ Root an. cond. $0.756444$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 + 2·2-s − 5·4-s + 2·7-s − 20·8-s − 3·9-s + 20·11-s + 4·14-s + 5·16-s − 6·18-s + 40·22-s − 28·23-s + 2·25-s − 10·28-s − 76·29-s + 118·32-s + 15·36-s + 52·37-s + 52·43-s − 100·44-s − 56·46-s − 45·49-s + 4·50-s + 20·53-s − 40·56-s − 152·58-s − 6·63-s + 111·64-s + ⋯
 L(s)  = 1 + 2-s − 5/4·4-s + 2/7·7-s − 5/2·8-s − 1/3·9-s + 1.81·11-s + 2/7·14-s + 5/16·16-s − 1/3·18-s + 1.81·22-s − 1.21·23-s + 2/25·25-s − 0.357·28-s − 2.62·29-s + 3.68·32-s + 5/12·36-s + 1.40·37-s + 1.20·43-s − 2.27·44-s − 1.21·46-s − 0.918·49-s + 2/25·50-s + 0.377·53-s − 5/7·56-s − 2.62·58-s − 0.0952·63-s + 1.73·64-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$0.327422$$ Root analytic conductor: $$0.756444$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{21} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 441,\ (\ :1, 1),\ 1)$$

Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.9124476892$$ $$L(\frac12)$$ $$\approx$$ $$0.9124476892$$ $$L(2)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ $$1 + p T^{2}$$
7$C_2$ $$1 - 2 T + p^{2} T^{2}$$
good2$C_2$ $$( 1 - T + p^{2} T^{2} )^{2}$$
5$C_2^2$ $$1 - 2 T^{2} + p^{4} T^{4}$$
11$C_2$ $$( 1 - 10 T + p^{2} T^{2} )^{2}$$
13$C_2^2$ $$1 - 290 T^{2} + p^{4} T^{4}$$
17$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
19$C_2^2$ $$1 - 290 T^{2} + p^{4} T^{4}$$
23$C_2$ $$( 1 + 14 T + p^{2} T^{2} )^{2}$$
29$C_2$ $$( 1 + 38 T + p^{2} T^{2} )^{2}$$
31$C_2^2$ $$1 - 1154 T^{2} + p^{4} T^{4}$$
37$C_2$ $$( 1 - 26 T + p^{2} T^{2} )^{2}$$
41$C_2^2$ $$1 + 1438 T^{2} + p^{4} T^{4}$$
43$C_2$ $$( 1 - 26 T + p^{2} T^{2} )^{2}$$
47$C_2^2$ $$1 - 3650 T^{2} + p^{4} T^{4}$$
53$C_2$ $$( 1 - 10 T + p^{2} T^{2} )^{2}$$
59$C_2^2$ $$1 - 1154 T^{2} + p^{4} T^{4}$$
61$C_2^2$ $$1 - 6242 T^{2} + p^{4} T^{4}$$
67$C_2$ $$( 1 - 74 T + p^{2} T^{2} )^{2}$$
71$C_2$ $$( 1 + 62 T + p^{2} T^{2} )^{2}$$
73$C_2^2$ $$1 - 8930 T^{2} + p^{4} T^{4}$$
79$C_2$ $$( 1 + 46 T + p^{2} T^{2} )^{2}$$
83$C_2^2$ $$1 - 5666 T^{2} + p^{4} T^{4}$$
89$C_2^2$ $$1 - 14114 T^{2} + p^{4} T^{4}$$
97$C_2^2$ $$1 - 15746 T^{2} + p^{4} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$